Miftellanea Cnriofa. 337 



II. Let fuch ordinate d b, or (equal to it in 

 the Afymptote) AF, be To divided in L, M, N, 

 (by Perpendiculars cutting the Hyperbola in 

 /, m, w, &c.) as that FL, LM, M N, be as 



^— r, C^c That is, (b continually de- 



m 9 mm m 



creafing as that each Antecedent be to its Confe- 



quent,asi to — , or as m to i. See Fig. 5*. Tab. 

 rn 



1 3 . This is done by taking A F 9 A L, A N, 6c* 

 in fuch proportion. For, of continual Propor- 

 tionals, the Differences are aifo continually pro- 

 portional, and in the fame proportion. For let 

 A, B, C, D, &c. be fuch Proportionals, and their 

 Differences a, b, c, 6c. That is, A — B-=za, 

 B — C^b, C — D, =c, 6c. 



Then, becaufe A, B, C, D, £5c. are in conti- 

 nual proportion, 



That is, A. B : : B. C : : C. D : : 6c. 



And dividing A— B.B : : B— C.C : : C— D 

 D :: 6c 



That is, 4. B :: b.C:: d. D :.:.{&. 



And alternly a. b. c. 6c. :: B. CD. 6c. : : 



A. B. C. tfc. 

 That is, in continual proportion as A to B, 



or asm to 1. 



14. This being done ; the Hyperbolick Spaces 

 F /, Lm, Mw, &c. are equal. As is demon- 

 ftrated by Gregory San-Vincont j and as fuch is 

 commonly admitted. 



15. So that F/, Lm y Mn 9 6c, may fitly re- 

 prefent equal Times, in which are difpatched 

 unequal Lengths, reprefented by F L 9 L M, M N, 

 6c Z 16. And 



